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Properties

Label 2-2070-345.344-c1-0-44
Degree $2$
Conductor $2070$
Sign $0.979 + 0.200i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (2.15 + 0.610i)5-s + 4.98·7-s + 8-s + (2.15 + 0.610i)10-s − 2.47·11-s − 3.76i·13-s + 4.98·14-s + 16-s + 1.23i·17-s − 7.97i·19-s + (2.15 + 0.610i)20-s − 2.47·22-s + (2.46 − 4.11i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.962 + 0.272i)5-s + 1.88·7-s + 0.353·8-s + (0.680 + 0.193i)10-s − 0.747·11-s − 1.04i·13-s + 1.33·14-s + 0.250·16-s + 0.300i·17-s − 1.82i·19-s + (0.481 + 0.136i)20-s − 0.528·22-s + (0.514 − 0.857i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.979 + 0.200i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (2069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.979 + 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.983438348\)
\(L(\frac12)\) \(\approx\) \(3.983438348\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (-2.15 - 0.610i)T \)
23 \( 1 + (-2.46 + 4.11i)T \)
good7 \( 1 - 4.98T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 + 3.76iT - 13T^{2} \)
17 \( 1 - 1.23iT - 17T^{2} \)
19 \( 1 + 7.97iT - 19T^{2} \)
29 \( 1 - 6.53iT - 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 4.87iT - 41T^{2} \)
43 \( 1 + 3.98T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 2.28iT - 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 8.76T + 67T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 0.127iT - 79T^{2} \)
83 \( 1 - 3.92iT - 83T^{2} \)
89 \( 1 - 4.67T + 89T^{2} \)
97 \( 1 + 8.10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.812690550543312864542848445849, −8.421367659003401013425571947001, −7.33746106860715017072254173562, −6.80197105478281445552806472564, −5.47831626762573580400281449961, −5.22025080450794414112666178964, −4.52367125163197298743642633055, −3.04752014102643258301156411926, −2.32348556391091459944511775309, −1.27082232845834668569536452829, 1.69740895538533557525664659116, 1.87968909554009116890202135660, 3.37403234967802268571216292880, 4.60968715053558388666580395258, 5.01528473147234580086163913532, 5.76236366644286801071675115176, 6.60813970367525168125012389784, 7.75008189227254699673829238952, 8.138655366978820163840458957072, 9.134160866720912905584625949933

Graph of the $Z$-function along the critical line